Finite Group Databasebeta

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#72224570

order7776
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Splittings

Fitting, socle, SylowC35 ⋊ C2×SD16

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, cosocleC3×#66472C23
Fitting, socle, hypocenterC35C2×SD16
prime radical#72214794C2
LCSG ▷ C3×#66472 ▷ C3×C3⋊(C3⋊(C3⋊D3)) ▷ C35
UCSC1

862 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 862 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 3 generators on 21 points:

(1,2,3)(4,5,6,9,8,12,7,11)(13,16,17,21,15,18,14,19);(2,3)(5,6,9,11)(7,8,12,10)(14,15,18,20)(16,17,21,19);(14,18)(15,20)(16,21)(17,19)

Polycyclic presentation:

2^5,3^5|b^2=e,c^2=d,d^2=e,c^b=cd,d^b=de,f^b=f^2,g^a=g^2,g^b=h,g^c=h,g^d=gh^2,g^e=g^2,h^a=h^2,h^b=g^2,h^c=gh^2,h^d=g^2h^2,h^e=h^2,i^a=gi,i^b=ij,i^c=ij,i^d=j^2,i^e=i^2,j^a=g^2hj,j^b=ij^2,j^c=i^2j,j^d=i,j^e=j^2

Conjugacy classes & character table

51 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×#72210346, #72259512, #72259580
maximal normal subgroup ofD3×#72210346, #72259512, #72259580

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 3, 3, 3, 3]);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 2, F.8);;
gap> SetConjugate(coll, 7, 3, F.8);;
gap> SetConjugate(coll, 7, 4, F.7*F.8^2);;
gap> SetConjugate(coll, 7, 5, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> SetConjugate(coll, 8, 2, F.7^2);;
gap> SetConjugate(coll, 8, 3, F.7*F.8^2);;
gap> SetConjugate(coll, 8, 4, F.7^2*F.8^2);;
gap> SetConjugate(coll, 8, 5, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.7*F.9);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.10^2);;
gap> SetConjugate(coll, 9, 5, F.9^2);;
gap> SetConjugate(coll, 10, 1, F.7^2*F.8*F.10);;
gap> SetConjugate(coll, 10, 2, F.9*F.10^2);;
gap> SetConjugate(coll, 10, 3, F.9^2*F.10);;
gap> SetConjugate(coll, 10, 4, F.9);;
gap> SetConjugate(coll, 10, 5, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);